# If A and B are two events such that

Question:

If $A$ and $B$ are two events such that $A \subset B$ and $P(B) \neq 0$, then which of the following is correct?

A. $P(A \mid B)=\frac{P(B)}{P(A)}$

B. $\mathrm{P}(\mathrm{A} \mid \mathrm{B})<\mathrm{P}(\mathrm{A})$

C. $P(A \mid B) \geq P(A)$

D. None of these

Solution:

If $A \subset B$, then $A \cap B=A$

$\Rightarrow P(A \cap B)=P(A)$

Also, $P(A) Consider$P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A)}{P(B)} \neq \frac{P(B)}{P(A)} \ldots$(1) Consider$\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{\mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{B})}$...(2) It is known that, P (B) ≤ 1$\Rightarrow \frac{1}{P(B)} \geq 1\Rightarrow \frac{P(A)}{P(B)} \geq P(A)$From$(2)$, we obtain$\Rightarrow \mathrm{P}(\mathrm{A} \mid \mathrm{B}) \geq \mathrm{P}(\mathrm{A})$...(3)$\therefore \mathrm{P}(\mathrm{A} \mid \mathrm{B})$is not less than$\mathrm{P}(\mathrm{A})\$

Thus, from (3), it can be concluded that the relation given in alternative C is correct.